Appendix to V. Mathai and J. Rosenberg's paper "A noncommutative sigma-model"

TL;DR

This paper proves a conjecture regarding the minimal energy of unitaries in noncommutative tori and establishes lower bounds for energies of unital *-endomorphisms, advancing understanding in noncommutative geometry.

Contribution

It confirms Rosenberg's conjecture on minimal energies and provides new lower bounds for energies of *-endomorphisms in noncommutative tori.

Findings

Proved Rosenberg's conjecture on minimal energy values.

Established lower bounds for energies of unital *-endomorphisms.

Enhanced understanding of energy minimization in noncommutative tori.

Abstract

We prove a conjecture of Rosenberg about the minimal value for energies of untaries in the two-dimensional noncommutative tori and answer a question of his about lower bounds for energies of unital *-endomorphisms of the two-dimensional noncommutative tori.

Equations

$$E(u)=\frac{1}{2}\mathrm{tr}({\delta }_1(u{)}^{\ast }{\delta }_1(u)+{\delta }_2(u{)}^{\ast }{\delta }_2(u))$$

$$E(u)=\frac{1}{2}(\Vert {\delta }_1(u){\Vert }_2^2+\Vert {\delta }_2(u){\Vert }_2^2).$$

$$\displaystyle{\rm tr}(a_{j})={\rm tr}((U^{m}V^{n})^{*}\delta_{j}(U^{m}V^{n}))=\left\{\begin{array}[]{lr}2\pi im&\mbox{ if }j=1;\\ 2\pi in&\mbox{ if }j=2.\end{array}\right.$$

$$\Vert {\delta }_j(u){\Vert }_2^2$$

$$\mathrm{tr}(uv)=\mathrm{tr}(\lambda vu)=\lambda \mathrm{tr}(uv),$$

$$-\mathrm{tr}(u)\mathrm{tr}(v)$$

$$(2\pi {)}^2\Vert u-\mathrm{tr}(u){\Vert }_2^2$$

$$|\mathrm{tr}(u){|}^2=\Vert \mathrm{tr}(u){\Vert }_2^2=\Vert u{\Vert }_2^2-\Vert u-\mathrm{tr}(u){\Vert }_2^2\ge 1-\frac{1}{2{\pi }^2}E(u),$$

$$|\mathrm{tr}((u-\mathrm{tr}(u))v)|\le \Vert (u-\mathrm{tr}(u))v{\Vert }_2=\Vert u-\mathrm{tr}(u){\Vert }_2\le (\frac{1}{2{\pi }^2}E(u){)}^{1/2}.$$

$$(1-t)(1-s)\le |\mathrm{tr}(u)\mathrm{tr}(v){|}^2\le t.$$

$$t(1-w/2)\ge t(1-s)\ge 1-(t+s)=1-w,$$

$$w=t+s\ge \frac{1-w}{1-w/2}+\frac{w}{2}.$$

Full text

Partially supported by NSF Grant DMS-0701414.

This short note is an appendix to [6].

Let $\theta\in{\mathbb{R}}$. Denote by $A_{\theta}$ the rotation $C^{*}$-algebra generated by unitaries $U$ and $V$ subject to $UV=e^{2\pi i\theta}VU$, and by $A_{\theta}^{\infty}$ its canonical smooth subalgebra. Denote by ${\rm tr}$ the canonical faithful tracial state on $A_{\theta}$ determined by ${\rm tr}(U^{m}V^{n})=\delta_{m,0}\delta_{n,0}$ for all $m,n\in{\mathbb{Z}}$. Denote by $\delta_{1}$ and $\delta_{2}$ the unbounded closed $*$-derivations of $A_{\theta}$ defined on some dense subalgebras of $A_{\theta}$ and determined by $\delta_{1}(U)=2\pi iU$, $\delta_{1}(V)=0$, and $\delta_{2}(U)=0$, $\delta_{2}(V)=2\pi iV$. The energy [9], $E(u)$, of a unitary $u$ in $A_{\theta}$ is defined as

[TABLE]

when $u$ belongs to the domains of $\delta_{1}$ and $\delta_{2}$, and $\infty$ otherwise.

Rosenberg has the following conjecture [9, Conjecture 5.4].

For a $*$-endomorphism $\varphi$ of $A_{\theta}^{\infty}$, its energy [6], ${\mathcal{L}}(\varphi)$, is defined as $2E(\varphi(U))+2E(\varphi(V))$. Mathai and Rosenberg’s Conjecture 3.1 in [6] about the minimal value of ${\mathcal{L}}(\varphi)$ follows directly from Conjecture 1.

Denote by $H$ the Hilbert space associated to the GNS representation of $A_{\theta}$ for ${\rm tr}$, and denote by $\|\cdot\|_{2}$ its norm. We shall identify $A_{\theta}$ as a subspace of $H$ as usual. Then (1) can be rewritten as

[TABLE]

Now we prove Conjecture 1, and hence also prove Conjecture 3.1 of [6].

[TABLE]

We have

[TABLE]

and “=” holds if and only if $a_{j}={\rm tr}(a_{j})$. It follows that $E(u)\geq 2\pi^{2}(m^{2}+n^{2})$, and “=” holds if and only if $\delta_{1}(u)=2\pi imu$ and $\delta_{2}(u)=2\pi inu$. Now the theorem follows from the fact that the elements $a$ in $A_{\theta}$ satisfying $\delta_{1}(a)=2\pi ima$ and $\delta_{2}(a)=2\pi ina$ are exactly the scalar multiples of $U^{m}V^{n}$. ∎

When $\theta\in{\mathbb{R}}$ is irrational, the $C^{*}$-algebra $A_{\theta}$ is simple [10, Theorem 3.7], has real rank zero [1, Theorem 1.5], and is an $A{\mathbb{T}}$-algebra [5, Theorem 4]. It is a result of Elliott that for any pair of $A{\mathbb{T}}$-algebras with real rank zero, every homomorphism between their graded $K$-groups preserving the graded dimension range is induced by a $*$-homomorphism between them [4, Theorem 7.3]. The graded dimension range of a unital simple $A{\mathbb{T}}$-algebra $A$ is the subset $\{(g_{0},g_{1})\in K_{0}(A)\oplus K_{1}(A):0\lvertneqq g_{0}\leq[1_{A}]_{0}\}\cup(0,0)$ of the graded $K$-group $K_{0}(A)\oplus K_{1}(A)$ [8, page 51]. It follows that, when $\theta$ is irrational, for any group endomorphism $\psi$ of $K_{1}(A_{\theta})$, there is a unital $*$-endomorphism $\varphi$ of $A_{\theta}$ inducing $\psi$ on $K_{1}(A_{\theta})$. It is an open question when one can choose $\varphi$ to be smooth in the sense of preserving $A_{\theta}^{\infty}$, though it was shown in [2, 3] that if $\theta$ is irrational and $\varphi$ restricts to a $*$-automorphism of $A_{\theta}^{\infty}$, then $\psi$ must be an automorphism of the rank-two free abelian group $K_{1}(A_{\theta})$ with determinant $1$. When $\psi$ is the zero endomorphism, from Theorem 2 one might guess that ${\mathcal{L}}(\varphi)$ could be arbitrarily small. It is somehow surprising, as we show now, that in fact there is a common positive lower bound for ${\mathcal{L}}(\varphi)$ for all $0<\theta<1$. This answers a question Rosenberg raised at the Noncommutative Geometry workshop at Oberwolfach in September 2009.

Theorem 3 is a direct consequence of the following lemma.

[TABLE]

and hence ${\rm tr}(uv)=0$. Thus

[TABLE]

We may assume that both $u$ and $v$ belong to the domains of $\delta_{1}$ and $\delta_{2}$. For any $m,n\in{\mathbb{Z}}$, denote by $a_{m,n}$ the Fourier coefficient $\left<u,U^{m}V^{n}\right>$ of $u$. Then $a_{0,0}={\rm tr}(u)$, and

[TABLE]

Thus

[TABLE]

and

[TABLE]

Similarly, $|{\rm tr}(v)|^{2}\geq 1-\frac{1}{2\pi^{2}}E(v)$.

Write $\frac{1}{2\pi^{2}}E(u)$ and $\frac{1}{2\pi^{2}}E(v)$ as $t$ and $s$ respectively. We just need to show that $t+s\geq 3-\sqrt{5}$. If $t\geq 1$ or $s\geq 1$, then this is trivial. Thus we may assume that $1-t,1-s>0$. Then

[TABLE]

Equivalently, $t(1-s)\geq 1-(t+s)$. Without of loss generality, we may assume $s\geq t$. Write $t+s$ as $w$. Then

[TABLE]

and hence

[TABLE]

It follows that $w^{2}-6w+4\leq 0$. Thus $w\geq 3-\sqrt{5}$. ∎